This paper addresses the problem of secure consensus tracking control for nonlinear leader-follower multi-agent systems, with a specific focus on safeguarding followers’ private information from external network eavesdroppers or internal untrusted neighbors. To tackle this problem, we first employ the dynamic linearization approximation technique to transform the nonlinear system models into equivalent linear forms that involve unknown time-varying pseudopartial derivatives. Then, a novel model-free secure distributed adaptive control (MFSDAC) framework is proposed using an encoding-decoding mechanism and Paillier encryption. Within this framework, we develop a secure distributed control scheme using a recursive form and design an adaptive updating law with a modified projection to estimate the time-varying pseudoparial derivatives. To enhance security, we introduce an adjustable parameter and a random integer into the distributed communication protocol, effectively preventing the disclosure of followers’ data during both network transmissions and controller evaluations. Additionally, parameter selection rules for the controller, quantizer, and adaptive updating law are provided, along with convergence analysis and guarantees against quantizer saturation. Finally, numerical simulations confirm that the proposed MFSDAC framework successfully achieves leader-following tracking and secure data transmission, even in the presence of external network eavesdroppers or internal untrusted neighbors. Note to Practitioners—Multi-agent systems (MASs) provide a versatile framework for modeling and understanding various real-world applications, including autonomous systems, traffic management, and distributed sensor networks. Designing effective control strategies for MASs is essential to enhance cooperation and coordination among agents, strengthen system-level resilience and adaptability, and tackle complex tasks that surpass the capabilities of individual agents. A key challenge in this area is ensuring network security and protecting individual privacy, especially in the presence of external eavesdroppers and untrusted internal neighbors. To tackle this challenge, we propose a model-free secure distributed adaptive control framework for nonlinear leader-follower MASs. This framework incorporates a confidential communication protocol that leverages homomorphic encryption to protect sensitive information. The proposed framework has undergone rigorous stability analysis and been validated through numerical simulations, demonstrating its feasibility and effectiveness in achieving secure consensus control of nonlinear MASs.

This correction addresses a citation oversight in the paper[1]. While[1]has cited recent publication about the model-free adaptive control (MFAC) theory, it omits foundational references that are critical to understanding the core principles underlying MFAC, such as dynamic linearization, generalized Lipschitz condition, and the time-varying pseudopartial derivative. To rectify this, the correction supplements[1]with citations to pioneering contributions in MFAC theory. This ensures proper attribution and provides readers with a deeper and more comprehensive understanding of the theoretical foundations of MFAC.

Homomorphic encryption enables secure control of networked systems with untrusted computing entities but greatly increases communication overhead compared with plaintext-based control. To address this, we propose a dynamic mode-switching secure control framework that alternates between plaintext and encrypted operations. In plaintext mode, sensor measurements are obfuscated using random dithered quantization, while in encrypted mode, measurements are fully encrypted to provide enhanced system security. To evaluate the framework, a worst-case eavesdropping scenario is introduced, where the adversary has complete knowledge of the system model and access to all plant-controller communications. Within this setting, we develop three switching strategies—periodic, random, and error-based—to govern the operational mode. Rigorous theoretical analysis establishes formal guarantees for both control performance and security, alongside deriving a critical parameter condition for decryption correctness. Ensuring the signal-to-noise ratio of the eavesdropper’s estimate remains below 10 dB, simulations show that periodic and random switching reduce communication by at least 30%. Error-based switching with an appropriate threshold (β = 1 × 10−4) achieves more than 70% reduction. These results confirm that the proposed framework effectively balances control performance, system security, and communication overhead, rendering it well-suited for resource-constrained networked systems.

This paper addresses the issue of relative position tracking control for spacecraft rendezvous and proximity operations with a freely tumbling target, considering spatial motion constraints, including path and relative velocity limits, as well as mass uncertainty and external disturbances. To handle these challenges, a two-stage operation is proposed to ensure compliance with path constraints. In each stage, the constrained zones are parameterized using geometric features of the space, and corresponding artificial potential functions (APFs) are designed. Based on these APFs, an adaptive tracking controller is developed inside a backstepping control frame that allows the pursuer to reach terminal docking position while respecting the spatial motion constraints. Lyapunov stability analysis shows that the proposed control scheme ensures asymptotic convergence of tracking errors, provided that the APFs avoid local minima. Finally, numerical simulations are performed to validate the efficacy of the proposed control scheme.

This paper addresses the secure control problem of uncertain networked control systems (NCSs) with an untrusted controller, such as a cloud-based controller. Using the Paillier cryptosystem, we propose a model-based adaptive encrypted networked control framework that incorporates joint static and dynamic quantization policies, and an encrypted controller. Within this framework, a novel adaptive updating law is designed to compensate for the nonlinearities caused by quantization and parameter estimation, contributing to the asymptotic convergence of the system state. Moreover, since no raw information is transmitted over the communication network or utilized inside the controller device, the security of NCSs is effectively ensured. Further, to tackle the issue of ciphertext expansion, we develop an event-based adaptive encrypted control scheme. This scheme employs two distinct triggering functions, which significantly reduces frequent information transmission and skillfully eliminates the comprehensive nonlinearities arising from quantization, estimation, and triggering. It is worth noting that, due to the integer-based nature of the Paillier cryptosystem, both the system state and the estimated controller gain should be quantized. This differs significantly from existing works on quantized-based adaptive control methods, where only a single variable, either the system state or the control input, is quantized. Finally, simulation results validate the effectiveness of the proposed encrypted control frameworks.

This article addresses the event-triggered optimal tracking control problem for leader-follower spacecraft formation flying system using the adaptive dynamic programming technique. In order to solve the Hamilton–Jacobi–Bellman equation, a single-critic neural network (NN) is developed to approximate the optimal cost function. Moreover, by combining the parameter projection rule and gradient descent algorithm, a semiglobal adaptive update law is derived to tune the critic NN. In doing so, a continuous near optimal tracking controller is presented. Subsequently, an input-state-dependent event-triggered mechanism is designed to ensure that the near optimal tracking controller is implemented only when specific events occur, which significantly reduces the execution frequency of the control command. Remarkably, benefiting from the construction of an input-based triggering error, the conventional assumption on the Lipschitz continuity of the controller is tactfully removed, thus erasing the computable demand on the unknown Lipschitz constants. Rigorous analysis on the system stability and Zeno-free behavior is provided successively. Finally, numerical simulations on two formation satellites in low Earth orbit validate the effectiveness of the theoretical scheme.

This article addresses the event-triggered coordinated control problem for multiple Euler-Lagrange systems subject to parameter uncertainties and external disturbances. Based on the event-triggered technique, a distributed coordinated control scheme is first proposed, where the neural network-based estimation method is incorporated to compensate for parameter uncertainties. Then, an input-based continuous event-triggered (CET) mechanism is developed to schedule the triggering instants, which ensures that the control command is activated only when some specific events occur. After that, by analyzing the possible finite-time escape behavior of the triggering function, the real-time data sampling and event monitoring requirement in the CET strategy is tactfully ruled out, and the CET policy is further transformed into a periodic event-triggered (PET) one. In doing so, each agent only needs to monitor the triggering function at the preset periodic sampling instants, and accordingly, frequent control updating is further relieved. Besides, a parameter selection criterion is provided to specify the relationship between the control performance and the sampling period. Finally, a numerical example of attitude synchronization for multiple satellites is performed to show the effectiveness and superiority of the proposed coordinated control scheme.

This article considers the connectivity preservation coordinated control problem for multiple spacecraft systems subject to limited communication resources and sensing capability. By constructing a novel bump function, a distance-dependent dynamic communication network model is first presented, which characterizes the interaction strength as a nonlinear smooth function varying with the relative distance of spacecraft continuously. Subsequently, based on an edge-tension potential function, a distributed event-driven coordinated control scheme is proposed to achieve formation consensus, while ensuring that adjacent spacecraft is always within the allowable connectivity range. Meanwhile, to avoid redundant data transmissions, a hybrid dynamic event-triggered mechanism with maximum triggering interval is developed to schedule the communication frequency among spacecraft. It is proven that the onboard communication resources occupation can be reduced significantly and the Zeno phenomenon is strictly excluded. Finally, the efficiency of the proposed method for, as an example, four-spacecraft formation system is substantiated.

This paper addresses the relative position coordinated control problem for spacecraft formation flying under an undirected communication graph, whilst considering mass uncertainties, external disturbances, and limited communication resources. A new event-triggered information transmission mechanism is first presented, where each spacecraft only requires accessing to the states of neighbors intermittently. Subsequently, a novel event-based coordinated control scheme is proposed by combining a smooth adaptive projection rule that confines the parameter estimations to well-defined bounded convex hypercubes. Under the proposed control framework, the information exchange among spacecraft occurs only when the specified event is triggered, thereby significantly reducing the communication load and saving the onboard resources. Furthermore, a positive lower bound on interevent time intervals is guaranteed to exclude Zeno behavior. By virtue of Lyapunov stability analysis and graph theory, it is proved that the relative position tracking errors can converge to small invariant sets around the origin, and that all closed-loop signals are bounded, even in the presence of mass uncertainties and external disturbances. Finally, numerical simulations are given to evaluate the effectiveness and highlight the advantages of the developed control algorithm.